A strongly regular graph has constant degree and fixed common-neighbour counts for adjacent and nonadjacent vertex pairs.
If a strongly regular graph on vertices has degree , with common neighbours for adjacent vertices and for nonadjacent vertices, then its adjacency matrix satisfies
A connected regular graph is strongly regular exactly when its adjacency matrix has three distinct eigenvalues, apart from the complete graphs, which have two. The forward direction follows from the adjacency-matrix relation for a strongly regular graph; conversely, the quadratic polynomial vanishing on the two nonconstant eigenspaces is a scalar multiple of the all-ones matrix.
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A strongly regular graph is a specific type of graph characterized by a regular structure that satisfies certain conditions regarding its vertices and edges. Formally, a strongly regular graph \( G \) is defined by three parameters \( (n, k, \lambda, \mu) \) where: - \( n \) is the total number of vertices in the graph.